Set is Subset of Lower Closure

Theorem

Let $\struct {S, \preceq}$ be an ordered set.

Let $X$ be a subset of $S$.

Then $X \subseteq X^\preceq$

where $X^\preceq$ denotes the lower closure of $X$.

Proof

Let $x \in X$.

By definition of reflexivity:

$x \preceq x$

Thus by definition of lower closure:

$x \in X^\preceq$

$\blacksquare$

Sources

• Mizar article WAYBEL_0:16